Lesson 3-3

Rates of Change in the Natural and Social Sciences

 

Objectives

• Understand the mathematical modeling process of derivatives (rates of changes) in the real world

Vocabulary

• Mathematical Model – an equation that models a process (usually in the real world)

Example 1A particle is moving along an axis so that at time t its position is f(t) = t³ - 6t² + 6 feet. What is the velocity at time t? 

 What is the velocity at 3 seconds? 

 Is the particle moving left or right at 3 seconds?

v(t) = f’(t) = 3t² - 12t ft/s

v(3) = f’(3) = 3(3)² - 12(3) = 27 – 36 = -9 ft/s

Since v(3) = -9 < 0 then particle is moving left

Example 2A stone is thrown upward from a 70 meter cliff so that its height above ground is f(t) = 70 + 3t - t².

What is the velocity of the stone as it hits the ground?

v(t) = f’(t) = 3 - 2t

0 = f(t) = 70 + 3t – t²

0 = (10 – t)(7 + t) so t = 10 when stone hits ground

v(10) = f’(10) = 3 – 2(10) = -17 m/s

Example 3A particle moves according to the position function, s(t) = t³ - 9t² + 15t + 10, t ≥ 0 where t is in seconds and s(t) is in feet. Find the velocity at time t.   

When is the particle at rest?   

v(t) = f’(t) = 3t² - 18t + 15 ft/s

v(t) = f’(t) = 3t² - 18t + 15 ft/s 0 = 3t² - 18t + 15 0 = 3(t – 5)(t – 1) so vel = 0 at t = 1 and 5

Example 3 contA particle moves according to the position function, s(t) = t³ - 9t² + 15t + 10, t≥0 where t is in seconds and s(t) is in feet. When is the particle moving to the right?  Find the total distance traveled in the first 8 seconds.   Draw a diagram to illustrate the particle’s motion.

When v(t) > 0 so when t < 1 and t > 5

Add |distance| in these intervals: 0<t<1, 1<t<5, 5<t<8s(1)-s(0) + s(5)-s(1) + s(8)-s(5) = (17-10) + (-15-17) + (66- -15)7 + |32| + 81 = 120

10 17-15 66

Example 4Water is flowing out of a water tower in such a way that after t minutes there are 10,000 – 10t – t³ gallons remaining. How fast is the water flowing after 2 minutes?

f(t) = v’(t) = -10 – 3t² gal/sec

f(2) = -10 – 3(2)² = -10 – 12 = -22 gal / sec

Example 5A space shuttle is 16t + t³ meters from its launch pad t seconds after liftoff.

What is its velocity after 3 seconds?

f(t) = v’(t) = 16 + 3t² m/sec

f(3) = 16 + 3(3)² = 16 + 27 = 43 m/sec

Example 6The numbers of yellow perch in a heavily fished portion of Lake Michigan have been declining rapidly. Using the data below, estimate the rate of decline in 1996 by averaging the slopes of two secant lines.

t , years 1993 1994 1995 1996 1997 1998P(t) , population

(millions)4.2 4.0 3.7 3.6 3.3 3.1

m97-95 = (3.3 – 3.7) / 2 = - 0.2

m98-94 = (3.1 – 4.0) / 2 = - 0.45

t ≈ m98-94 = (-0.2 + .45) / 2 = - 0.325

Summary & Homework

• Summary:– Derivatives can model real world rates of

change

• Homework: – pg: 208-210: 8,9,10